Verification for this problem is provided by comparing the values of significant problem variables with the values produced by an equivalent model in ABAQUS/Standard. The ABAQUS/Standard analyses use 5-point Simpson integration only and a HAFTOL value of 1.0 × 10³. The ABAQUS/Explicit analyses are run with 5-point Simpson integration and 3-point Gauss integration. The rigid surface is modeled as analytical and acts as a pure master surface in the ABAQUS/Standard analysis. The contact constraints account for the shell thickness in the ABAQUS/Explicit analyses only. The ABAQUS/Explicit results shown below are for an element-based rigid surface with kinematic enforcement of contact constraints, except where noted otherwise.

Table 1.6.10–1 and Table 1.6.10–2 compare tip displacements, tip velocities, and whole model energies at several points along the beam's symmetry axis. Tip displacements and velocities are averaged over the four nodes at the tip of the beam. The results from the ABAQUS/Explicit analyses using Simpson (5-point) and Gauss (3-point) integration through the thickness of the shell demonstrate slight sensitivity of the response to the choice of the integration rule. Corresponding components of displacement and velocity at the tip of the beam are within 0.1% and 0.5%, respectively, for the ABAQUS/Explicit (Simpson integration) and ABAQUS/Standard analyses without the cylinder. For the problem with the cylinder, the significant components of displacement and velocity are within 2% and 8%, respectively, between the ABAQUS/Explicit and ABAQUS/Standard results with Simpson integration.

Figure 1.6.10–3 shows contours of equivalent plastic strain on the bottom surface of the beam for the ABAQUS/Explicit analysis using Simpson integration without the rigid cylinder. Figure 1.6.10–4 shows the corresponding plot for the ABAQUS/Standard analysis. The contours are plotted on the deformed shapes of the beam. After 0.08 seconds a plastic hinge has formed at the fixed end of the beam for both cases.

Figure 1.6.10–5 and Figure 1.6.10–6 show contours of equivalent plastic strain on the bottom surface of the beam impacting the rigid cylinder for the ABAQUS/Explicit analysis with Simpson integration and the ABAQUS/Standard analysis, respectively.

Figure 1.6.10–7 through Figure 1.6.10–10 show the final configuration near the rigid cylinder for four ABAQUS/Explicit analyses. Figure 1.6.10–7 corresponds to an analysis with an analytical rigid surface and kinematic contact. Figure 1.6.10–8 corresponds to an analysis with an analytical rigid surface and penalty contact. In both of these cases the analytical surface is the pure master surface of the contact pair. Contact is enforced at the slave nodes accounting for the shell thickness, and there is some penetration of the rigid surface into the shell. The final position of the tip is slightly different in Figure 1.6.10–7 and Figure 1.6.10–8, which is attributable to impacts being perfectly plastic with kinematic contact and elastic with penalty contact (see “Contact formulation for ABAQUS/Explicit contact pairs,” Section 29.4.4 of the ABAQUS Analysis User's Manual). Figure 1.6.10–9 corresponds to an analysis with an element-based rigid surface and kinematic contact. Figure 1.6.10–10 corresponds to an analysis with an element-based rigid surface and penalty contact. Penetration of the rigid surface into the shell surface is repelled only in Figure 1.6.10–10, because this is the only case in which the rigid surface nodes are weighted at all as slave nodes.